Galilean contractions of $W$-algebras

Abstract

Infinite-dimensional Galilean conformal algebras can be constructed by contracting pairs of symmetry algebras in conformal field theory, such as $W$-algebras. Known examples include contractions of pairs of the Virasoro algebra, its $N=1$ superconformal extension, or the $W_3$ algebra. Here, we introduce a contraction prescription of the corresponding operator-product algebras, or equivalently, a prescription for contracting tensor products of vertex algebras. With this, we work out the Galilean conformal algebras arising from contractions of $N=2$ and $N=4$ superconformal algebras as well as of the $W$-algebras $W(2,4)$, $W(2,6)$, $W_4$, and $W_5$. The latter results provide evidence for the existence of a whole new class of $W$-algebras which we call Galilean $W$-algebras. We also apply the contraction prescription to affine Lie algebras and find that the ensuing Galilean affine algebras admit a Sugawara construction. The corresponding central charge is level-independent and given by twice the dimension of the underlying finite-dimensional Lie algebra. Finally, applications of our results to the characterisation of structure constants in $W$-algebras are proposed.

arXiv comment45 pages, v2: minor changes, references added

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Cited by in corpus (11)

• The (super)conformal BMS${}_3$ algebra
• Self-dual and logarithmic representations of the twisted Heisenberg--Virasoro algebra at level zero
• CFT correlators, $\mathcal{W}$-algebras and Generalized Catalan Numbers
• On conformal field theories based on Takiff superalgebras
• Higher-order Galilean contractions
• The BRST quantisation of chiral BMS-like field theories
• Multi-graded Galilean conformal algebras
• Remarks on BMS${}_3$ invariant field theories: correlation functions and nonunitary CFTs
• Asymmetric Galilean conformal algebras
• Galilean $W_3$ algebra
• Free-field construction of Carrollian $W_N$-algebras